Set \(\Lambda := \lbrace \alpha : K_\alpha \mbox{ \ has an identity element}\rbrace \) . If \(|\Lambda | \le 1\) , set
\( K := \lbrace  a_1 a_2 \ldots , a_n : n\in \mathbb {N}, a_1 a_2\ldots a_n \mbox{ \ is a reduced word on } \lbrace K_\alpha \rbrace \rbrace  .\) 
Note that if \(|\Lambda |=1\) , then we treat the identity as any other arbitrary element in that semihypergroup. For convenience of notations, we rename the idenity element as \(x_e\)  in this case, so that it can be included in finite sequences of reduced words.

If \(|\Lambda |>1\) , let \(e\)  denote the empty word on \(\lbrace K_\alpha \rbrace \)  and set \(e= 1_{K_\alpha }\)  for each \(\alpha \in \Lambda \)  where \(1_{K_\alpha }\)  denotes the identity element of \(K_\alpha \) . Now set
\( K := \lbrace e\rbrace  \ \cup \ \lbrace  a_1 a_2\ldots a_n : n\in \mathbb {N}, \ a_1 a_2\ldots a_n \mbox{ is a reduced word on } \lbrace K_\alpha \rbrace \rbrace  .\) 
For any \(x_1 x_2\ldots x_n\) , \(y_1 y_2\ldots y_m \in K\)  where \(x_i \in K_{\alpha _i}, y_j \in K_{\beta _j}\)  for \(i=1, 2, \ldots , n\) ; \(j=1, 2, \ldots , m\)  and any subset \(A \subset K_{\alpha _0}\)  where \(\alpha _0 \ne \alpha _n , \beta _1\)  we define that:
\((x_1 \ldots x_n) A (y_1 \ldots y_m) := {\left\lbrace \begin{array}{ll}\lbrace x_1 x_2 \ldots x_n a y_1 y_2 \ldots y_m : a\in A\rbrace , & \hspace{-8.67204pt} \mbox{ if } \ e \in \hspace{-8.67204pt} / \ A .\\& \\\lbrace (x_1 x_2 \ldots x_n y_1 y_2 \ldots y_m)\rbrace  \cap K & \\\cup \lbrace (x_1 x_2 \ldots x_n a y_1 y_2 \ldots y_m) : a\in A, a\ne e\rbrace , & \hspace{-8.67204pt} \mbox{ if } \ e \in A .\\\end{array}\right.}\) 
Now for any two sets \(A_1\subset K_{\alpha _1}\) , \(A_2\subset K_{\alpha _2}\)  where \(\alpha _1 \ne \alpha _2\)  define
\(A_1A_2 &:=& \lbrace a_1 a_2: a_1 \in A_1, a_2 \in A_2\rbrace , \mbox{ if } e\in \hspace{-7.94974pt} / \ A_1, A_2 . \\A_1A_2 &:=& \lbrace a_1 a_2: a_1 \in A_1\setminus \lbrace e\rbrace , a_2 \in A_2\rbrace  \cup \lbrace a_2 : a_2 \in A_2\rbrace , \mbox{ if } e \in A_1, e \in \hspace{-7.94974pt} / \ A_2 .\\A_1A_2 &:=& \lbrace a_1 a_2: a_1 \in A_1, a_2 \in A_2 \setminus \lbrace e\rbrace \rbrace  \cup \lbrace a_1 : a_1 \in A_1\rbrace , \mbox{ if } e \in \hspace{-7.94974pt} / \ A_1, e \in A_2 .\\A_1A_2 &:=& \lbrace a_1 a_2: a_1 \in A_1 \setminus \lbrace e\rbrace , a_2 \in A_2 \setminus \lbrace e\rbrace \rbrace  \cup \lbrace a_1 : a_1 \in A_1 \setminus \lbrace e\rbrace \rbrace \\ && \cup \lbrace a_2 : a_2 \in A_2 \setminus \lbrace e\rbrace \rbrace  \cup \lbrace e\rbrace , \mbox{ if } e \in A_1, A_2 .\) 
Similarly for \(A_1\subset K_{\alpha _1}\) , \(A_2\subset K_{\alpha _2}\) , \(A_3\subset K_{\alpha _3}\)  where \(\alpha _1 \ne \alpha _2 \ne \alpha _3\) , we define
\(A_1A_2A_3 &:=& \lbrace a_1 a_2 a_3: a_1 \in A_1, a_2 \in A_2, a_3 \in A_3\rbrace , \mbox{ if } e\in \hspace{-7.94974pt} / \ A_1, A_2, A_3 .\\A_1A_2A_3 &:= & \lbrace a_1 a_2 a_3: a_1 \in A_1\setminus \lbrace e\rbrace , a_2 \in A_2, a_3 \in A_3 \rbrace \\&& \cup \lbrace a_2 a_3 : a_2 \in A_2, a_3\in A_3\rbrace , \mbox{ if } e \in A_1, \ e \in \hspace{-7.94974pt} / \ A_2, A_3 .\\A_1A_2A_3 &:=& \lbrace a_1 a_2 a_3: a_1 \in A_1\setminus \lbrace e\rbrace , a_2 \in A_2 \setminus \lbrace e\rbrace , a_3 \in A_3\rbrace  \\&& \cup \lbrace a_2 a_3 : a_2 \in A_2\setminus \lbrace e\rbrace , a_3\in A_3\rbrace  \\&& \cup \lbrace a_1 a_3: a_1 \in A_1\setminus \lbrace e\rbrace , a_3\in A_3\rbrace  \cup \lbrace a_3: a_3\in A_3\rbrace , \\&& \mbox{ if } e \in A_1, A_2, \ e \in \hspace{-7.94974pt} / \ A_3 .\\A_1A_2A_3 &:=& \lbrace a_1 a_2 a_3: a_1 \in A_1 \setminus \lbrace e\rbrace , a_2 \in A_2 \setminus \lbrace e\rbrace , a_3\in A_3\setminus \lbrace e\rbrace \rbrace  \\&& \cup \lbrace a_1 a_2 : a_1 \in A_1 \setminus \lbrace e\rbrace , a_2\in A_2\setminus \lbrace e\rbrace \rbrace  \cup \lbrace a_2, a_3 : a_2 \in A_2 \setminus \lbrace e\rbrace ,\\&& a_3\in A_3\setminus \lbrace e\rbrace \rbrace  \cup \lbrace a_1, a_3 : a_1 \in A_1 \setminus \lbrace e\rbrace , a_3\in A_3\setminus \lbrace e\rbrace \rbrace  \\&& \cup \lbrace a_1: a_1\in A_1\setminus \lbrace e\rbrace \rbrace  \cup \lbrace a_2: a_2\in A_2\setminus \lbrace e\rbrace \rbrace  \\&& \cup \lbrace a_3: a_3\in A_3\setminus \lbrace e\rbrace \rbrace  \cup \lbrace e\rbrace , \mbox{ if } e \in A_1, A_2, A_3 .\) 
We can get the other cases following the same pattern.
In general for any \(n \in \mathbb {N}\) , consider subsets \(A_i \subset K_{\alpha _i}\)  for \(i=1, 2, \ldots , n\) , where \(\alpha _i \ne \alpha _{i+1}\)  for \(i= 1, 2, \ldots , (n-1)\) . Let \(e \in A_{i_k}\)  for \(k= 1, 2, \ldots , m\)  where \(i_{k_1}<i_{k_2}\)  for \(k_1<k_2\)  and \(e \in \hspace{-7.94974pt} / \ A_i\)  whenever \(i \ne i_k\)  for any \(k= 1, 2, \ldots , m\) .

Then continuing in the same manner as above, we get that
\( A_1A_2\ldots A_n := {\left\lbrace \begin{array}{ll}\lbrace  a_1 a_2\ldots a_n \in K : a_i \in A_i, i = 1, 2, \ldots , n\rbrace  & \\\cup \ ( F_n \cap K ) \cup \lbrace e\rbrace  \ , & \mbox{ if } m=n \\&\\\lbrace  a_1 a_2\ldots a_n \in K : a_i \in A_i, i = 1, 2, \ldots , n\rbrace  &\\\cup \ ( F_n \cap K ) \ , & \mbox{ if } m\ne n\\\end{array}\right.}\) 
where \(F_n\)  is defined as the following:
\(\begin{split}F_n := & \bigcup _{l=1}^m \bigcup _{\small {\begin{array}{c}\hspace{3.61371pt}1\le k_1<k_2<\ldots \\ \ldots <k_l\le m\end{array}}} \hspace{-3.61371pt} \Big {\lbrace  } (a_1 a_2 \ldots a_{{i_{k_1}-1}} a_{{i_{k_1}}+1} \ldots a_{i_{k_2}-1} a_{{i_{k_2}}+1} \ldots a_{i_{k_l}-1} a_{i_{k_l}+1} \ldots \\[-3.5ex]& \hspace{151.76744pt} \ldots a_{n-1}a_n) : a_j \in A_j, j= 1, 2, \ldots , n \Big {\rbrace } .\end{split}\) 
Note that from now on unless otherwise mentioned, whenever we write a product \(A_1A_2\ldots A_n\)  of subsets \(A_i \subset K_{\alpha _i}\) , we assume that \(\alpha _i \ne \alpha _{i+1}\)  for \(i=1, 2, \ldots , (n-1)\) . Also it follows immediately from the above construction that for any such sets \(A_1, A_2, \ldots , A_n\)  we have
\( A_1A_2\ldots A_n = A_1(A_2A_3\ldots A_n) = A_1(A_2A_3\ldots A_{n-1})A_n = (A_1A_2\ldots A_{n-1})A_n .\) 
Natural Topology on \(K\) :
 
Now we first introduce a certain topology \(\tau \)  on \(K\) . Afterwards we will define a binary operation `\(*\) ' on the measure space \(M(K)\)  and then investigate how the operation gives rise to an associative algebra on \(M(K)\)  and how the Borel sets act under the said operation and the cone topology on \(M^+(K)\)  induced by \(\tau \) . First, we set
\(\mathcal {B}:= \lbrace U_1U_2, \ldots , U_n : U_i \mbox{ is an open subset of } K_{\alpha _i}, i=1,2, \ldots n; n\in \mathbb {N}\rbrace  .\) 
We need to show that \(\mathcal {B}\)  serves as a base for a topology on \(K\) . Pick any element \(x_1 x_2\ldots x_n \in K\setminus \lbrace e\rbrace \)  where \(x_i \in K_{\alpha _i}\) , \(i=1,2, \ldots , n\) . Since \(x_1 x_2\ldots x_n\)  is a reduced word, \(x_i \ne 1_{K_{\alpha _i}}\) , even if \(\alpha _i \in \Lambda \) . Thus we can find an open neighbourhood \(V_{x_i}\)  of \(x_i\)  in \(K_{\alpha _i}\)  such that \(e\in \hspace{-7.94974pt} / \ V_{x_i}\) . Then \(x_1 x_2\ldots x_n \in V_{x_1}V_{x_2}\ldots V_{x_n} \in \mathcal {B} \ \) .

Also note that if \(e\in K\) , then for any open neighborhood \(U\)  of \(e=1_{K_\alpha }\)  in \(K_\alpha \)  where \(\alpha \in \Lambda \) , we have that \(e\in U \in \mathcal {B}\) . Now let \(U_1U_2\ldots U_n, V_1V_2\ldots V_m \in \mathcal {B}\)  where \(U_i\subset K_{\alpha _i}\)  and \(V_j \subset K_{\beta _j}\)  for \(i=1, 2, \ldots n\) ; \(j=1, 2, \ldots , m\) . Pick any element \(x_1 x_2 \ldots x_l \in U_1U_2\ldots U_n \cap V_1V_2\ldots V_m\) . Note that \(l \le \min ({n, m})\)  and hence for each \(k=1, 2, \ldots l\) , there exists some \(i_k \in \lbrace 1, 2, \ldots , n\rbrace \)  and \(j_k \in \lbrace 1, 2, \ldots , m\rbrace \)  such that \(i_1 < i_2 < \ldots < i_l\) ,  \(j_1 < j_2 < \ldots < j_l\)  and \(x_k \in U_{i_k} \cap V_{j_k}\)  for each \(k=1, 2, \ldots , l \ \) .

Set \(W_k:=U_{i_k} \cap V_{j_k}\)  for each \(k=1, 2, \ldots , l\) . Since each \(W_k\)  is open in \(K_{\alpha _{i_k}}=K_{\beta _{j_k}}\)  and \(x_k \ne e\)  for each \(k\) , we have that \(x_1 x_2 \ldots x_l \in W_1W_2\ldots W_l \in \mathcal {B}\) . Now to show that \(W_1W_2\ldots W_l \subset U_1U_2\ldots U_n \cap V_1V_2\ldots V_m\) , pick any \(y_1 y_2 \ldots y_t \in W_1W_2\ldots W_l\) . Then for \(k = 1, 2, \ldots t\) , we have \(y_k \in W_{s_k} = U_{i_{s_k}}\cap V_{j_{s_k}}\) , for some \(s_k\in \lbrace 1, 2, \ldots , l\rbrace  \ \) .

Since \(x_1 x_2 \ldots x_l \in U_1U_2\ldots U_n\)  we must have that \(e\in U_i\)  whenever \(i\in \lbrace 1,2, \ldots n\rbrace  \setminus \lbrace i_k : k= 1, 2, \ldots , l\rbrace \) . Again since \(y_1 y_2\ldots y_t \in W_1W_2\ldots W_l\) , we must have that \(e\in W_{s} = U_{i_{s}}\cap V_{j_{s}}\subset U_{i_s}\)  whenever \(s\in \lbrace 1, 2, \ldots , l\rbrace \setminus \lbrace s_k: k=1, 2, \ldots , t\rbrace \) . Combining these two statements gives us that \(e\in U_i\)  whenever \(i\in \lbrace 1, 2, \ldots , n\rbrace  \setminus \lbrace i_{s_k}: k=1, 2, \ldots , t\rbrace \) . Also, we know that \(y_k \in U_{i_{s_k}}\)  for each \(k= 1, 2, \ldots , t\) . Hence it follows that \(y_1 y_2 \ldots y_t \in U_1U_2\ldots U_n \ \) .

Similarly since \(x_1 x_2 \ldots x_l \in V_1V_2\ldots V_m\) , we must have that \(e\in V_j\)  whenever \(j\in \lbrace 1,2, \ldots m\rbrace  \setminus \lbrace j_k : k= 1, 2, \ldots , l\rbrace \)  and since \(y_1 y_2 \ldots y_t \in W_1W_2\ldots W_l\) , we must have that \(e\in W_{s} = U_{i_{s}}\cap V_{j_{s}}\subset V_{j_s}\)  whenever \(s\in \lbrace 1, 2, \ldots , l\rbrace \setminus \lbrace s_k: k=1, 2, \ldots , t\rbrace \) , we can proceed in the same manner to get that \(y_1 y_2 \ldots y_t \in V_1V_2\ldots V_m \ \) .

Thus finally we see that \(x_1 x_2 \ldots x_l \in W_1W_2\ldots W_l \subset \big {(}U_1U_2\ldots U_n \cap V_1V_2\ldots V_m\big {)}\)  where \(W_1W_2\ldots W_l \in \mathcal {B}\) , implying that \(\mathcal {B}\)  serves as a base for a topology on \(K\) . We denote the topology on \(K\)  generated by \(\mathcal {B}\)  as \(\tau \) .


Natural Binary Operation \(*\)  on \(M(K)\) :
 
As pointed out in the first section of this text, in order to define a binary operation on \(M(K)\)  it is sufficient to define a binary operation on the point-mass measures on \(M(K)\) . First of all, if \(\Lambda \ne \varnothing \)  then for any \(x_1 x_2\ldots x_n\in K\)  we define
\(p_e*p_e=p_e \mbox{ \ \ ; \ \ } \ p_e*p_{x_1 x_2\ldots x_n} = p_{x_1 x_2\ldots x_n} = p_{x_1 x_2\ldots x_n}*p_e .\) 
Now for any two elements \(x=x_1 x_2\ldots x_n, y=y_1 y_2\ldots y_m \in K\setminus \lbrace e\rbrace \)  where
\(x_i \in K_{\alpha _i}, y_j \in K_{\beta _j}\)  for \(i=1, 2, \ldots , n\) ;  \(j=1, 2, \ldots , m\)  and for any Borel set \(E^{\prime } \subseteq K\)  we define:
\((p_x*p_y) (E^{\prime }) = {\left\lbrace \begin{array}{ll}p_{(x_1 \ldots x_n y_1 \ldots y_m)}(E^{\prime }) , & \hspace{-3.61371pt} \mbox{ if } \alpha _n \ne \beta _1.\\& \hspace{-3.61371pt} \\(p_{x_n}*p_{y_1})(E) ,& \hspace{-3.61371pt} \mbox{ if } \alpha _n = \beta _1, \alpha _{n-1}\ne \beta _2\\& \hspace{-3.61371pt} \mbox{ and } E^{\prime } \cap \big {(}(x_1 \ldots x_{n-1})K_{\alpha _n}(y_{2} \ldots y_m)\big {)}\\& \hspace{-3.61371pt} \ = (x_1 \ldots x_{n-1})E(y_{2} \ldots y_m).\\& \hspace{-3.61371pt} \\(p_{x_n}*p_{y_1})(E\cup \lbrace e\rbrace ) , & \hspace{-3.61371pt} \mbox{ if } \alpha _n = \beta _1, \alpha _{n-1}=\beta _2\\& \hspace{-3.61371pt} \mbox{ and } E^{\prime } \cap \big {(}(x_1 \ldots x_{n-1})K_{\alpha _n}(y_{2} \ldots y_m)\big {)}\\& \hspace{-3.61371pt} \ = (x_1 \ldots x_{n-1})E(y_{2} \ldots y_m) \ne \varnothing .\\& \hspace{-3.61371pt} \\0 , & \hspace{-3.61371pt} \mbox{ if } \alpha _n = \beta _1, \alpha _{n-1}=\beta _2\\& \hspace{-3.61371pt} \mbox{ and } E^{\prime } \cap \big {(}(x_1 \ldots x_{n-1})K_{\alpha _n}(y_{2} \ldots y_m)\big {)}\\& \hspace{-3.61371pt} \ =\varnothing .\\\end{array}\right.}\) 
Note that when \(\alpha _n = \beta _1\) , then \((p_x*p_y)\)  mimics the behaviour of the probability measure \((p_{x_{n}}*p_{y_{1}})\)  and hence is a probability measure with support \((x_1 \ldots x_{n-1})\mbox{ supp}(p_{x_{n}}*p_{y_{1}})(y_{2} \ldots y_m)\) .

\((p_x*p_y)\)  has compact support:
 
Due to the construction of \(\tau \) , for any two closed subsets \(C_1 \subset K_{\alpha _1}\)  and \(C_2 \subset K_{\alpha _2}\)  where \(\alpha _1\ne \alpha _2\) , we have that \(C_1C_2\)  is a closed subset of \(K\) . This is true since \(K_{\alpha _1} K_{\alpha _2}\)  is open in \(K\)  and we have that
\(K_{\alpha _1}K_{\alpha _2} \setminus C_1C_2 = (K_{\alpha _1}\setminus C_1).K_{\alpha _2} \cup K_{\alpha _1}.(K_{\alpha _2}\setminus C_2),\) 
which is open in \(K_{\alpha _1} K_{\alpha _2}\) . Similarly, we can show that \(C_1C_2\ldots C_n\)  is a closed set in \(K\)  for any \(n\in \mathbb {N}\)  where \(C_i \in K_{\alpha _i}\)  for \(1\le i \le n\)  and \(\alpha _i \ne \alpha _{i+1}\)  for \(1\le i \le n-1\) . In fact, it is now easy to see that the set \(\lbrace C_1C_2\ldots C_n : n\in \mathbb {N}, C_i \mbox{ \ is a closed set in some \ } K_{\alpha _i}\rbrace \)  is the family of basic closed sets in \(K\) .

Now let both \(C_1\)  and \(C_2\)  are compact subsets of \(K_{\alpha _1}\)  and \(K_{\alpha _2}\)  respectively and let \(\lbrace A_\gamma \rbrace _{\gamma \in I}\)  be a family of closed subsets of \(C_1C_2\)  with finite intersection property. Then we must have that \(A_\gamma = B_1^\gamma B_2^\gamma \)  for some closed subsets \(B_i^\gamma \subset C_i \subset K_{\alpha _i}\)  for each \(\gamma \) , \(i=1, 2\) . Now for any \(\lbrace \gamma _1, \gamma _2, \ldots , \gamma _n\rbrace  \subset I\)  we have that \(\cap _1^n B_1^{\gamma _i} B_2^{\gamma _i} = \cap _1^n A_{\gamma _i} \ne \varnothing \) .

Hence in particular, \(\cap _1^n B_1^{\gamma _i} \ne \varnothing \)  for any such finite subset \(\lbrace \gamma _1, \gamma _2, \ldots , \gamma _n\rbrace  \subset I\) . So \(\lbrace B_1^\gamma \rbrace _{\gamma \in I}\)  is a family of subsets in \(C_1\)  with the finite intersection property. But \(C_1\)  is compact in \(K_{\alpha _1}\)  and hence there exists some \(x_1 \in C_1\)  such that \(x_1 \in \cap _{\gamma \in I} B_1^\gamma \) . Similarly, since \(C_2\)  is also compact in \(K_{\alpha _2}\) , there exists some \(x_2 \in C_2\)  such that \(x_2 \in \cap _{\gamma \in I} B_2^\gamma \) . Thus we see that there exists an element \(x \in C_1C_2\)  such that \(x\in \cap _{\gamma \in I} A_\gamma \)  where \(x\)  is given as the following:
\(x = {\left\lbrace \begin{array}{ll}(x_1x_2) , & \mbox{ if } x_1, x_2 \ne e .\\(x_1) , & \mbox{ if } x_1\ne e, x_2 = e .\\(x_2) , & \mbox{ if } x_1 =e, x_2 \ne e .\\e , & \mbox{ if } x_1= x_2 = e .\end{array}\right.}\) 
Thus we see that \(C_1C_2\)  has to be compact too. Now let \(C_1,C_2, C_3\)  be compact subsets of \(K_{\alpha _1},K_{\alpha _2}, K_{\alpha _3}\)  respectively. Note that if \(\alpha _i \ne \alpha _j\)  for any \(i, j \in \lbrace 1, 2, 3\rbrace \)  or if \(e\notin \ \cap _{i=1}^3 C_i\)  then we can proceed similarly as above to show that \(C_1C_2C_3\)  gives us a compact set in \(K\) . Otherwise, first consider the following case.

Let \(C_0:=(a_1\ldots a_s)C(b_1 \ldots b_t)\) , where \(C\)  is a compact set in \(K_{\alpha _0}\)  and \( (a_1a_2\ldots a_s),\\ (b_1 b_2 \ldots b_t) \in K \setminus \lbrace e\rbrace \)  where \(a_i\in K_{\gamma _i}\)  and \(b_j\in K_{\delta _j}\)  for \(1\le i \le s, 1\le j \le t\)  such that \(\alpha _0 \ne \gamma _s, \delta _1\) . We claim that any such \(C_0\)  is compact in \((K, \tau )\) .

Let \(\lbrace U_\alpha \rbrace \)  be any basic open cover of \(C_0\) . Hence \(U_\alpha = U_1^\alpha U_2^\alpha \ldots U_{n_\alpha }^\alpha \)  for some \(n_\alpha \in \mathbb {N}\)  where each \(U_k^\alpha \)  is an open set in some \(K_\alpha \)  such that the product exists. Now for each \(x \in C\setminus \lbrace e\rbrace \)  there exists \(\alpha _x\)  such that \((a_1\ldots a_s x b_1 \ldots b_t) \in U_{\alpha _x}\) . Therefore there exists \(i_x\in \lbrace 1, 2, \ldots , n_{\alpha _x}\rbrace \)  such that \(x\in U_{i_x}^{\alpha _x}\)  where \(U_{i_x}^{\alpha _x}\)  is an open set in \(K_{\alpha _0}\) . Thus \(\mathcal {U}:= \lbrace U_{i_x}^{\alpha _x}\rbrace _{x\in C}\)  serves an open cover of the set \(C\setminus \lbrace e\rbrace \) .

If \(e \notin C\) , then \(\mathcal {U}\)  serves as an open cover of \(C\)  and hence has a finite subcover \(\lbrace U_1, U_2, \ldots , U_p\rbrace \)  where \(U_k = U_{i_{x_k}}^{\alpha _{x_k}}\)  for some \(x_k \in C\)  for \(1\le k \le p\) . Thus \(\lbrace U_{\alpha _{x_1}}, U_{\alpha _{x_2}}, \ldots , U_{\alpha _{x_p}}\rbrace \)  serves as a finite subcover of \(\lbrace U_\alpha \rbrace \) .

Now let \(e\in C\) . Pick \(\alpha ^{\prime } \in \Lambda \)  such that \(\alpha ^{\prime } \ne \alpha _0\) . Since \(K_{\alpha ^{\prime }}\)  is locally compact, we can find a compact neighborhood \(F\)  of \(e\)  in \(K_{\alpha ^{\prime }}\) . Let \(\mathcal {V}:=\lbrace V_\beta \rbrace \)  be an open cover of \(F\) . Then \(C\cup F\)  is a compact set in \(K_{\alpha _0}\cup K_{\alpha ^{\prime }}\)  with open cover \(\mathcal {U} \cup \mathcal {V}\) . Thus we get a finite subcover of the form \(\lbrace U_1, U_2, \ldots , U_p, V_{\beta _1}, V_{\beta _2}, \ldots V_{\beta _q}\rbrace \)  where \(U_k = U_{i_{x_k}}^{\alpha _{x_k}}\)  for some \(x_k \in C\setminus \lbrace e\rbrace \)  for \(1\le k \le p\) . Note that \(e\in V_{\beta _l}\)  for some \(l\in \lbrace 1, 2, \ldots , q\rbrace \)  here and hence \(\lbrace U_{i_{x_k}}\rbrace _{k=1}^p\)  covers \(C\setminus \lbrace e\rbrace \) .

Thus if \(\gamma _s=\delta _1\) , then \(\lbrace U_{\alpha _{x_1}}, U_{\alpha _{x_2}}, \ldots , U_{\alpha _{x_p}}\rbrace \)  covers \(C_0\) , as desired. Otherwise if \(\gamma _s\ne \delta _1\) , then there exists some \(\alpha ^{\prime }\)  such that \((a_1\ldots a_sb_1\ldots b_t) \in U_{\alpha ^{\prime }}\) . Hence in this case, \(\lbrace U_{\alpha _{x_1}}, U_{\alpha _{x_2}}, \ldots , U_{\alpha _{x_p}}, U_{\alpha ^{\prime }}\rbrace \)  serves as a finite subcover of \(C_0\) . Hence we see that any set of the form \(C_0\)  is compact in \((K, \tau )\) . Using similar argument we can immediately see that sets of the form \(C_1C_2C_3\)  are also compact where \(C_i \in K_{\alpha _i}\)  is compact.

In particular, for any two words \(x = (x_1x_2\ldots x_n), y = (y_1y_2\ldots y_m)\in K\)  where \(x_i \in K_{\alpha _i}\) , \(y_j \in K_{\beta _j}\)  and \(\alpha _n = \beta _1\) , we have that
\(\mbox{ supp}(p_x*p_y) = (x_1 \ldots x_{n-1})\mbox{ supp}(p_{x_{n}}*p_{y_{1}})(y_{2} \ldots y_m)\) 
is compact since \(\mbox{ supp}(p_{x_{n}}*p_{y_{1}})\)  is compact in \(K_{\alpha _{n}} = K_{\beta _{1}}\) . Thus it follows from the construction of convolution products that for any two words \(x, y \in K\)  we have that \((p_x*p_y)\)  is a probability measure with compact support. Also since convolution is associative in each \(M(K_\alpha )\)  and concatenation of finite words is associative, it follows from the construction that convolution products on \(M(K)\)  are associative.

\(\tau \)  is Hausdorff:
 
Now we proceed further to show that \((K, *)\)  indeed gives us a semihypergroup. First note that since each \(K_\alpha \)  is Hausdorff, for any two elements \(x_1 x_2\ldots x_n, y_1 y_2\ldots y_m \in K\)  where \(x_i \in K_{\alpha _i}\) , \(y_j \in K_{\beta _j}\) , we can find sets \(A_i\subset K_{\alpha _i}\) , \(B_j \subset K_{\beta _j}\)  for \(1\le i\le n\) , \(1\le j \le m\)  such that \(x_i \in A_i\) , \(y_j \in B_j\)  and \(A_i \cap B_j = \varnothing \)  for all \(i, j\) . Hence we have that \(x_1 x_2\ldots x_n\in A_1A_2\ldots A_n, \ y_1 y_2\ldots y_m\in B_1B_2\ldots B_m\)  where \(A_1A_2\ldots A_n \cap B_1B_2\ldots B_m = \varnothing \) . Also if \(e\in K\) , then for any such \(x_1 x_2\ldots x_n\)  since \(x_i \ne e\)  for any \(i\) , we can choose \(A_i\) 's such that \(e\in \hspace{-7.94974pt} / \ A_i\)  for any \(i\) . Then again using the fact that each \(K_\alpha \)  is Hausdorff, we can choose a neighborhood \(V\)  around \(e\)  such that \(V\cap A_i = \varnothing \)  for all \(i\)  and hence \(V \cap A_1A_2\ldots A_n = \varnothing \) . Thus we can easily see that the topology \(\tau \)  on \(K\)  is Hausdorff.

\(\tau \)  is locally compact:
 
Next pick any element \(x_1 x_2\ldots x_n \in K\)  where \(x_i \in K_{\alpha _i}\)  for \(1\le i\le n\) . Since each \(K_{\alpha _i}\)  is locally compact and Hausdorff, we can find a compact neighborhood \(C_i\)  of \(x_i\)  in \(K_{\alpha _i}\)  such that \(e \in \hspace{-7.94974pt} / \ C_i\) . Let \(U_i\)  be an open set in \(K_{\alpha _i}\)  such that \(C_i \subset U_i\)  and \(e \in \hspace{-7.94974pt} / \ U_i\) . Now consider the map \(\phi : U_1 \times U_2 \times \ldots \times U_n \rightarrow U_1U_2\ldots U_n\)  given by
\( \phi ((a_1, a_2, \ldots , a_n)) := (a_1a_2\ldots a_n) \) 
for any \((a_1, a_2, \ldots , a_n) \in U_1 \times U_2 \times \ldots \times U_n \ \) .

Note that since \(e \in \hspace{-7.94974pt} / \ U_i\)  for any \(i\) , the map \(\phi \)  is bijective. Moreover, for any basic open set \(V_1V_2\ldots V_n \subset U_1U_1\ldots U_n\)  we have \(\phi ^{-1}(V_1V_2\ldots V_n) = V_1\times V_2\times \ldots \times V_n\)  which is open in \(U_1 \times U_2 \times \ldots \times U_n\)  in the product topology. Hence \(\phi \)  is continuous, granting us that \(C_1C_2\ldots C_n = \phi (C_1\times C_2 \times \ldots \times C_n)\)  is a compact neighborhood of \(x_1x_2\ldots x_n\)  in \((K, \tau )\) . Also if \(e\in K\)  then there exists some \(\alpha _0\)  such that \(e\in K_{\alpha _0}\) . Since \(K_{\alpha _0}\)  is locally compact we can always find a compact neighborhood for \(e\) . Thus we see that \((K, \tau )\)  is a locally compact Hausdorff space.

Continuity of support with respect to Michael Topology:
 
Now in order to see if the map \((x, y) \mapsto supp (p_x*p_y) : K\times K \rightarrow \mathfrak {C}(K)\)  is continuous, first recall [1]} that the map \(\pi : X \rightarrow M(X)\)  given by \(x \mapsto p_x\)  is a homemorphism onto its image for any locally compact Hausdorff space \(X\) . Now let \(\lbrace (x_\alpha , y_\alpha )\rbrace \)  be a net in \(K\times K\)  that converges to some \((x, y)\)  where \(x= x_1x_2\ldots x_n\) , \(y=y_1y_2\ldots y_m \in K\)  such that \(x_i \in K_{\gamma _i}\)  and \(y_j \in K_{\delta _j}\)  for all \(i, j\) . Then \(\lbrace x_\alpha \rbrace \)  and \(\lbrace y_\alpha \rbrace \)  converges to \(x\)  and \(y\)  respectively. Then for each \(\alpha \)  we have that \(x_\alpha = x_1^\alpha x_2^\alpha \ldots x_{n_\alpha }^\alpha \)  for some \(n_\alpha \in \mathbb {N}\)  where \(x_i^\alpha \in K_{\alpha _i}\)  for \(1 \le i \le n_\alpha \) . But \(x_\alpha \)  eventually lies in any basic open set of the form \(U_1U_2\ldots U_n\)  around \(x\)  where \(U_i\)  is an open neighborhood of \(x_i\)  in \(K_{\gamma _i}\)  that does not contain \(e\) . Hence without loss of generality for each \(\alpha \)  we can assume that \(n_\alpha =n\)  and \({\alpha _i} = \gamma _i\)  for \(1\le i \le n\) . Similarly we can assume that \(y_\alpha = y_1^\alpha y_2^\alpha \ldots y_m^\alpha \)  where \(y_j^\alpha \in K_{\delta _j}\)  for \(1\le j \le m\) .

Now consider the Michael topology on \(\mathfrak {C}(K)\)  and recall [2]} that the map
\(x\mapsto \lbrace x\rbrace  : K \rightarrow \mathfrak {C}(K)\) 
is a homeomorphism onto its image which is also closed in \(\mathfrak {C}(K)\) . If \(\gamma _n \ne \delta _1\)  then using the same technique as above, we can easily see that the net \(\lbrace (x_1^\alpha \ldots x_n^\alpha y_1^\alpha \ldots \\ \ldots y_m^\alpha )\rbrace _\alpha \)  converges to \((x_1\ldots x_ny_1\ldots y_m)\)  in \(K\) . Hence using the homeomorphism mentioned above we have that \(supp (p_{x_\alpha }*p_{y_\alpha }) = \lbrace (x_1^\alpha \ldots x_n^\alpha y_1^\alpha \ldots y_m^\alpha )\rbrace \)  converges to \(\lbrace (x_1\ldots x_ny_1\ldots y_m)\rbrace  = supp (p_x*p_y)\)  in \(\mathfrak {C}(K)\)  in Michael Topology.

Now let \(\gamma _n = \delta _1\) . For any \(i_0 \in \lbrace 1, 2, \ldots n\rbrace \) , pick and fix open neighborhoods \(U_i\)  around \(x_i\)  in \(K_{\gamma _i}\)  for all \(i\ne i_0\) . Then for any open neighborhood \(V\)  around \(x_{i_0}\)  in \(K_{\gamma _{i_0}}\)  we have that \(x_1^\alpha x_2^\alpha \ldots x_{n}^\alpha \)  lie eventually in \(U_1U_2\ldots U_{i_0-1}VU_{i_0+1}\ldots U_n\) . Hence we must have that \(\lbrace x_{i_0}^\alpha \rbrace \)  lie eventually in \(V\) , i.e, \(\lbrace x_{i_0}^\alpha \rbrace \)  converges to \(x_{i_0}\)  in \(K_{\gamma _{i_0}}\) . Thus \(\lbrace x_i^\alpha \rbrace \)  converges to \(x_i\)  in \(K_{\gamma _i}\)  and \(\lbrace y_j^\alpha \rbrace \)  converges to \(y_j\)  in \(K_{\delta _j}\)  for each \(i, j\) . In particular, we have that the net \(\lbrace (x_n^\alpha , y_1^\alpha )\rbrace _\alpha \)  converges to \((x_n, y_1)\)  in \(K_{\gamma _n} \times K_{\gamma _n}\) . Since the map \((x, y) \mapsto supp (p_x*p_y) : K_{\gamma _n}\times K_{\gamma _n} \rightarrow \mathfrak {C}(K_{\gamma _n})\)  is continuous, we have that \(supp(p_{x_n^\alpha }* p_{y_1^\alpha })\)  converges to \(supp(p_{x_n}*p_{y_1})\)  in \(\mathfrak {C}(K_{\gamma _n})\) .

As discussed before, we also have that the nets \(\big {\lbrace }\lbrace x_i^\alpha \rbrace \big {\rbrace }_\alpha \)  and \(\big {\lbrace }\lbrace y_j^\alpha \rbrace \big {\rbrace }_\alpha \)  converge to \(\lbrace x_i\rbrace \)  and \(\lbrace y_j\rbrace \)  in \(\mathfrak {C}(K_{\gamma _i})\)  and \(\mathfrak {C}(K_{\delta _j})\)  respectively for \(1\le i < n\)  and \(1<j \le m\) . Hence it follows from the construction of basic open sets of \(K\)  that \(supp (p_{x_\alpha }*p_{y_\alpha }) = \lbrace (x_1^\alpha \ldots x_{n-1}^\alpha )\rbrace  supp (p_{x_n^\alpha }* p_{y_1^\alpha }) \lbrace (y_2^\alpha \ldots y_m^\alpha )\rbrace \)  converges to \(\lbrace (x_1\ldots x_{n-1})\rbrace  supp (p_{x_n}* p_{y_1}) \lbrace (y_2\ldots y_m)\rbrace  = supp (p_x *p_y)\)  in \(\mathfrak {C}(K)\) .

Finally, it follows immediately from the construction of convolution products that the map \(*: M(K) \times M(K) \rightarrow M(K)\)  is positive bilinear. Using the same technique as above we can see that the restricted map \(*|_{M^+(K) \times M^+(K)} : M^+(K) \times M^+(K) \rightarrow M^+(K)\)  is continuous. Thus we have that the pair \((K, *)\)  indeed forms a semihypergroup.

\(K_\alpha \)  as a sub-semihypergroup:
 
Note that for each \(\alpha \)  the map \(i_\alpha : K_\alpha \rightarrow K\)  given by
\(i_\alpha (x) = {\left\lbrace \begin{array}{ll}(x) , & \mbox{ if } x\ne e \\e , & \mbox{ if } x = e\end{array}\right.}\) 
where \((x)\)  is a word of length 1 in \(K\) , enables us to view \(K_\alpha \)  as a sub-semihypergroup of \(K\)  since \(i_\alpha (K_\alpha )*i_\alpha (K_\alpha ) \subset i_\alpha (K_\alpha )\)  and \(i_\alpha \)  is a homeomorphism onto its image.


Universal property:
 
Now let \(H\)  be a semihypergroup and \(\phi _\alpha : K_\alpha \rightarrow H\)  be a homomorphism for each \(\alpha \) . For each \(x=x_1x_2\ldots x_n \in K\)  where \(x_i \in K_{\alpha _i}\)  for \(1\le i \le n\)  define
\(\Gamma (p_x) := p_{\phi _{\alpha _1}(x_1)} * p_{\phi _{\alpha _2}(x_2)} * \ldots * p_{\phi _{\alpha _n}(x_n)} . \) 
We can then extend \(\Gamma \)  to \(M(K)\)  as in the proof of Theorem REF  to get a positive continuous linear map \(\Gamma : M(K) \rightarrow M(H)\) .

Since for any \(x\in K_\alpha \)  we have that \(\Gamma (p_(x)) = p_{\phi _\alpha (x)}\) , it follows immediately from Theorem REF  that \(\Gamma |_{M(K_\alpha )}= \Gamma \circ \Gamma _{i_\alpha }= \Gamma _{\phi _\alpha }\)  for each \(\alpha \) . Now to show that \(\Gamma \)  is a homomorphism, pick any two words \(x=x_1x_2\ldots x_n\) , \(y=y_1y_2\ldots y_m\)  in \(K\)  where \(x_i \in K_{\alpha _i}\) , \(y_j \in K_{\beta _j}\)  for \(1\le i \le n\) , \(1\le j \le m\) . Note that if \(\alpha _n \ne \beta _1\)  then we have that
\(\Gamma (p_x*p_y) &=& \Gamma (p_{x_1\ldots x_ny_1\ldots y_m})\\&=& p_{\phi _{\alpha _1}(x_1)}* \ldots *p_{\phi _{\alpha _n}(x_n)}*p_{\phi _{\beta _1}(y_1)}*\ldots * p_{\phi _{\beta _m}(y_m)}\\&=& \big {(}p_{\phi _{\alpha _1}(x_1)}* \ldots *p_{\phi _{\alpha _n}(x_n)}\big {)}*\big {(}p_{\phi _{\beta _1}(y_1)}*\ldots * p_{\phi _{\beta _m}(y_m)}\big {)}\\&=& \Gamma (p_{x_1x_2\ldots x_n}) * \Gamma (p_{y_1y_2\ldots y_m}) = \Gamma (p_x) * \Gamma (p_y)\) 
as required.
Now pick any \(x \in K_\gamma \)  and \(y, z \in K_\beta \)  where \(\gamma \ne \beta \) . Then there exists a net \(\lbrace m_\alpha \rbrace \)  in \(M_F^+(K_\beta )\)  such that \(m_\alpha \)  converges to \((p_y*p_z)\)  in \(M^+(K_\beta )\) . Hence \((p_x*m_\alpha )\)  converges to \((p_x*(p_y*p_z))= (p_x*p_y*p_z)\)  in \(M^+(K)\) . Then for each \(\alpha \)  we have that \(m_\alpha = \sum _{i=1}^{n_\alpha } c_i^\alpha p_{x_i^\alpha }\)  for some \(n_\alpha \in \mathbb {N}\) , \(c_i^\alpha \in \mathbb {C}\)  and \(x_i^\alpha \in K_\beta \) . Since \(\Gamma \)  is continuous on \(M^+(K)\) , we have that
\(\Gamma (p_x*p_y*p_z) &=& \Gamma \big {(}\lim _\alpha (p_x *m_\alpha )\big {)}\\&=& \lim _\alpha \Gamma (p_x * m_\alpha )\\&=& \lim _\alpha \Gamma \Big {(}p_x * \sum _{i=1}^{n_\alpha } c_i^\alpha p_{x_i^\alpha }\Big {)}\\&=&\lim _\alpha \Gamma \Big {(} \sum _{i=1}^{n_\alpha } c_i^\alpha (p_x*p_{x_i^\alpha })\Big {)}\\&=& \lim _\alpha \sum _{i=1}^{n_\alpha } c_i^\alpha \ \Gamma \big {(}p_x*p_{x_i^\alpha }\big {)}\\&=& \lim _\alpha \sum _{i=1}^{n_\alpha }c_i^\alpha \Big {(}\Gamma (p_x)*\Gamma (p_{x_i^\alpha })\Big {)}\\&=& \lim _\alpha \Gamma (p_x)*\Big {(}\sum _{i=1}^{n_\alpha }c_i^\alpha \Gamma (p_{x_i^\alpha })\Big {)}\\&=&\lim _\alpha \Gamma (p_x) * \Gamma \Big {(} \sum _{i=1}^{n_\alpha }c_i^\alpha p_{x_i^\alpha }\Big {)}\\&=& \Gamma (p_x) * \lim _\alpha \Gamma (m_\alpha )= \Gamma (p_x) * \Gamma (p_y*p_z)\) 
where the fourth equality follows since the convolution \((\mu , \nu ) \mapsto \mu *\nu : M(K)\times M(K) \rightarrow M(K)\)  is bilinear, the fifth and eighth equality follows from the linearity of \(\Gamma \)  on \(M(K)\)  and finally the seventh equality follows since the map \((\mu , \nu ) \mapsto \mu *\nu : M(H)\times M(H) \rightarrow M(H)\)  is bilinear.
Similarly for any \(x, y \in K_\gamma \)  and \(z\in K_\beta \)  where \(\gamma \ne \beta \)  we have that
\(\Gamma (p_x*p_y*p_z) = \Gamma (p_x*p_y) * \Gamma (p_z) .\) 
Using these two equalities simultaneously for any two words \(x=x_1x_2\ldots x_n\) , \(y=y_1y_2\ldots y_m\)  in \(K\)  where \(x_i \in K_{\alpha _i}\) , \(y_j \in K_{\beta _j}\)  for \(1\le i \le n\) , \(1\le j \le m\)  and \(\alpha _n = \beta _1\) , we have that
\(\Gamma (p_x*p_y) &=& \Gamma (p_{x_1x_2\ldots x_{n-1}} *p_{x_n}*p_{y_1}*p_{y_2y_3\ldots y_m})\\&=& \Gamma (p_{x_1x_2\ldots x_{n-1}}) *\Gamma (p_{x_n}*p_{y_1})*\Gamma (p_{y_2y_3\ldots y_m})\\&=& \Gamma (p_{x_1x_2\ldots x_{n-1}}) *\Gamma _{\phi _{\alpha _n}}(p_{x_n}*p_{y_1})*\Gamma (p_{y_2y_3\ldots y_m})\\&=& \Gamma (p_{x_1\ldots x_{n-1}}) *\Big {(}\Gamma _{\phi _{\alpha _n}}(p_{x_n})*\Gamma _{\phi _{\alpha _n}}(p_{y_1})\Big {)}*\Gamma (p_{y_2\ldots y_m})\\&=& \Gamma (p_{x_1\ldots x_{n-1}})*\Big {(}p_{\phi _{\alpha _n}(x_n)} * p_{\phi _{\alpha _n}(y_1)}\Big {)}*\Gamma (p_{y_2\ldots y_m})\\&=& \Big {(}p_{\phi _{\alpha _1}(x_1)} * \ldots * p_{\phi _{\alpha _{n-1}}(x_{n-1})}\Big {)}* \Big {(}p_{\phi _{\alpha _n}(x_n)} *p_{\phi _{\beta _1}(y_1)}\Big {)}\\&& * \Big {(}p_{\phi _{\beta _2}(y_2)} * \ldots * p_{\phi _{\beta _m}(y_m)}\Big {)}\\&=& \Big {(}p_{\phi _{\alpha _1}(x_1)} * \ldots * p_{\phi _{\alpha _{n-1}}(x_{n-1})}* p_{\phi _{\alpha _n}(x_n)}\Big {)}\\&& *\Big {(}p_{\phi _{\beta _1}(y_1)}* p_{\phi _{\beta _2}(y_2)} * \ldots * p_{\phi _{\beta _m}(y_m)}\Big {)}\\&=& \Gamma \big {(}p_{x_1x_2\ldots x_n}\big {)} * \Gamma \big {(}p_{y_1y_2\ldots y_m}\big {)} \ = \ \Gamma (p_x)*\Gamma (p_y)\) 
as required. Note that here the fourth equality follows since \(\Gamma _{\phi _{\alpha _n}}: M(K_{\alpha _n}) \rightarrow M(H)\)  induced by \(\phi _{\alpha _n}\)  is a homomorphism and the fifth equality follows from the construction of \(\Gamma _{\phi _{\alpha _n}}\)  as shown in Theorem REF .

Finally, note that if \(\Theta : M(K) \rightarrow M(H)\)  is another positive linear continuous homomorphism such that \(\Theta |_{M(K_\alpha )} = \Gamma _{\phi _\alpha }\)  then for any \(x_1x_2\ldots x_n \in K\)  where \(x_i\in K_{\alpha _i}\)  for \(1\le i \le n\)  we have that
\(\Theta (p_{x_1x_2\ldots x_n})&=& \Theta (p_{x_1}*p_{x_2} * \ldots * p_{x_n})\\&=& \Theta (p_{x_1})*\Theta (p_{x_2})* \ldots *\Theta (p_{x_n})\\&=& \Theta |_{M(K_{\alpha _1})}(p_{x_1})*\Theta |_{M(K_{\alpha _2})}(p_{x_2})*\ldots * \Theta |_{M(K_{\alpha _n})}(p_{x_n})\\&=& \Gamma _{\phi _{\alpha _1}}(p_{x_1}) * \Gamma _{\phi _{\alpha _2}}(p_{x_2})* \ldots * \Gamma _{\phi _{\alpha _n}}(p_{x_n})\\&=& \Gamma |_{M(K_{\alpha _1})}(p_{x_1})*\Gamma |_{M(K_{\alpha _2})}(p_{x_2})*\ldots * \Gamma |_{M(K_{\alpha _n})}(p_{x_n})\\&=& \Gamma (p_{x_1}) *\Gamma (p_{x_2})*\ldots * \Gamma (p_{x_n})\\&=& \Gamma (p_{x_1}*p_{x_2} * \ldots * p_{x_n})= \Gamma (p_{x_1x_2\ldots x_n}) .\) 
Thus we see that the map \(\Gamma \)  constructed above is unique and hence \((K, *)\)  satisfies all the conditions of Definition REF  giving us that \(K= \prod ^*_\alpha K_\alpha \ \) .
\(\Box \)
Remark 3.7 Note that in classical literature, this natural construction fails to provide a free product of topological semigroups or groups. But in the case of pure semihypergroups, the natural construction suffices to provide a free product structure, i.e, in the most trivial case, when applied to a family of semigroups which are pure semihypergroups, the natural construction explained above results in a semihypergroup, rather than a semigroup.


Open Questions and Further Work
As discussed briefly in the introduction, the lack of prior extensive research since its inception and the significant examples available in coset and orbit spaces of locally compact groups, Lie groups and homogeneous spaces, makes way to a number of exciting new avenues of research on semihypergroups.

We have already discussed [3]} spaces of almost periodic and weakly almost periodic functions, amenability, ideals and homomorphisms on a semihypergroup. The following is some immediate potential problems related to this article that we are currently working on and/or intend to work on in near future.

Problem 1: Investigate the behavior of free products on semihypergroups for compact case, thereby exploring amenability properties of the resulting free product. While the compactness of the individual components make some of the deductions easier enabling us to drop some prior restrictions, as in most cases, the lack of an algebra in the underlying space prevents any direct extension of similar results [4]}, [5]} to this setup.

Problem 2: Generalize the construction of the above free product for amalgamated products as well as investigate the properties of the resulting semihypergroup, similar to the works in [5]}.

Problem 3: Amend and update the construction of the above free product for a family of hypergroups, such that the resulting free product becomes a hypergroup. This needs updating the construction of the natural topology \(\tau \)  so that it takes into account the involution of respective hypergroups, eventually resulting in a natural involution of the form \(x_1x_2\ldots x_{n-1}x_n \mapsto x_{n}^- x_{n-1}^-\ldots x_2^-x_1^- \) .

acknowledgements
The author would like to sincerely thank her doctoral thesis advisor Dr. Anthony To-Ming Lau, for suggesting the topic of this study and for the helpful discussions during the course of this work. She is also grateful to the referee for the valuable comments and suggestions.
